Continuity

Core Definition

A function $f(x)$ is continuous at a point $c$ if and only if three conditions are satisfied: (1) $f(c)$ is defined, (2) $\lim_{x \to c} f(x)$ exists, and (3) $\lim_{x \to c} f(x) = f(c)$ . If any of these conditions fail, the function is discontinuous at $c$ .

Concept Explanation

Continuity ensures a function has no sudden jumps, breaks, or asymptotic escapes. We classify discontinuities into:

Removable: A single missing point (hole) that can be redefined.
Jump: Left and right-hand limits exist but are unequal.
Infinite: The function approaches $\pm\infty$ (vertical asymptote).

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Used to verify that physical systems don't experience impossible sudden state changes. Civil engineers verify continuity in road transitions, mechanical engineers model structural joints to prevent stress failures, and aerospace engineers analyze discontinuous shockwaves in supersonic aerodynamics.

Example Problem

Find the value of $k$ that makes $f(x)$ continuous at $x = 2$ :

f(x) = \begin{cases} kx^2 & x \le 2 \\ 2x + k & x > 2 \end{cases}

Solution

For $f(x)$ to be continuous at $x=2$ , the left-hand limit, right-hand limit, and function value must all be equal.

Step 1: Find left-hand limit and function value

f(2) = \lim_{x \to 2^-} f(x) = k(2)^2 = 4k

Step 2: Find right-hand limit

\lim_{x \to 2^+} f(x) = 2(2) + k = 4 + k

Step 3: Set them equal and solve for $k$

4k = 4 + k \implies 3k = 4 \implies k = \frac{4}{3}

Connections & References

Parent Concepts: Limits
Sub-concepts: Differentiability
Course Links: Calc-1 Session 05